You are absolutely right. And also wrong. You are right, when something becomes cooler, heat is released from it. Your mistake is you are looking for your answer from the wrong perspective, I believe.
When you are dealing with calorimetry and heat of reaction, it can seem contradictory at first. You learn that an exothermic reaction is one in which heat leaves, and an endothermic reaction is one in which heat enters the system. Then you go to determine if a process is endo or exo, and it seems backwards. When measuring, say a chemical reaction in water, you are determining whether the change was exothermic or endothermic. It can seem like the answer is exothermic when heat leaves the water. And it is- for the water. You want to determine the change for the chemical reaction.
So, if water temperature decreases, then heat has left the water. For the water, this would be exothermic. But for the reaction, heat has entered that system. So the reaction is then, of course, endothermic.
Remember, you are determining heat of the system. Not heat of surroundings.
System = chemicals Surroundings = Water